Mott insulators with strong spin orbit coupling have become a testbed for exotic quantum phases, spin liquids and emergent Majorana matter. In this context we present results for the thermal conductivity of the Kitaev-Heisenberg model on ladders and the Kitaev model on honeycomb lattices. In the pure Kitaev limit, and in contrast to other integrable spin systems, the ladder represents a perfect heat insulator. This is shown to be a direct fingerprint of fractionalization into mobile Majorana matter and a static Z2 gauge field. We find a full suppression of the Drude weight and a pseudogap in the conductivity. With Heisenberg exchange, we find a crossover from a heat insulator to conductor, due to recombination of fractionalized spins into triplons. Increasing the dimension, and for the 2D honeycomb lattice, we show that very similar behavior occurs with however dissipative heat transport resulting in the thermodynamic limit. Our findings rest on several approaches comprising a mean-field theory, complete summation over all gauge sectors, exact diagonalization, and quantum typicality calculations.